MathDB

1

f(2xy) + f(f(x + y)) = xf(y) + yf(x) + f(x + y)

f : R \to R

such that

f(2xy) + f(f(x + y)) = xf(y) + yf(x) + f(x + y)

for all real numbers

x

and

y

.

2

1

12 students, n perfomances, 6 in each , largest n

Twelve students participated in a theater festival consisting of

n

different performances. Suppose there were six students in each performance, and each pair of performances had at most two students in common. Determine the largest possible value of

n

.

3

1

system 3x3 in N^3 with LCM a^2 + b^2 = n lcm(a, b) + n^2 PMO 2019

Find all triples

(a, b, c)

of positive integers such that

a^2 + b^2 = n\cdot lcm(a, b) + n^2

b^2 + c^2 = n \cdot lcm(b, c) + n^2

c^2 + a^2 = n \cdot lcm(c, a) + n^2

for some positive integer

n

.

4

1

concyclic from Philippines, equal angles and circumcircle related

In acute triangle

ABC

with

\angle BAC > \angle BCA

, let

P

be the point on side

BC

such that

\angle PAB = \angle BCA

. The circumcircle of triangle

AP B

meets side

AC

again at

Q

. Point

D

lies on segment

AP

such that

\angle QDC = \angle CAP

. Point

E

lies on line

BD

such that

CE = CD

. The circumcircle of triangle

CQE

meets segment

CD

again at

F

, and line

QF

meets side

BC

at

G

. Show that

B, D, F

, and

G

are concyclic

2019 Philippine MO

Subcontests

f(2xy) + f(f(x + y)) = xf(y) + yf(x) + f(x + y)

12 students, n perfomances, 6 in each , largest n

system 3x3 in N^3 with LCM a^2 + b^2 = n lcm(a, b) + n^2 PMO 2019

concyclic from Philippines, equal angles and circumcircle related